I'm trying to lower the action on the bass side of one of my non-Alembics. There is the proper amount of relief in the neck. When I lower the bridge significantly, I start to fret out the upper register. Could it be that the nut is a bit high? (The string to fret distance at the first fret seems a bit more than on my other guitars). Or is this more likely a fret issue?

I think it is the nut height. Because if I capo the first fret, the action is lower than I need and I'm not fretting out the upper register any more than without the capo. In fact, I can raise the bridge and still have lower action than without the capo.

Nut height only affects open string buzzing, if you are fretting out in the upper register most likely your bridge height is too low & your neck needs adjusting. You could have some high frets too that may need filing.

I know that open string buzzing (which I don't have) is caused by lack of proper "nut action", but I'm fairly certain that nut height impacts other aspects of playability. Given the neck angle on a guitar, it seems reasonable to conclude that trying to lower action in the mid portion of the neck to compensate for a nut that is too high will cause precisely the problem I'm experiencing, something supported by the fact that a capo eliminates the issue I'm having. As I mentioned, the relief is just right and if I raise the bridge up, I'm doing precisely the opposite of what I'm trying to achieve.

Remember, many different problems can cause high action on your guitar. Neck relief and saddle height are only two contributors to high action. One of the main causes of high action is an improperly set up nut. This is probably the most overlooked cause of high action and rightfully so. It is usually pretty obvious to beginners that their neck has excessive relief and saddle pieces are too high. A raised nut might be a little less obvious to a beginner. They just know the guitar is uncomfortable to play. The action at the nut affects the guitar completely different than the action at bridge. High action at the nut is particularly bad because it will force the entire guitar to play uncomfortably. Usually, the problem continues to worsen higher on the fretboard. Like any high action problems, intonation can be affected. High action at the nut can make the guitar untunable at the lower frets. Your open chords might not even sound right with an excessively high nut. Needless to say, the nut height plays an important roll in both the playability and functionality of your guitar. Lets talk a little about different styles of nuts before we get into how to lower them.